Re: Explanation of this hypothesis of SUBSTITUTION RULE requested


Arturo Magidin wrote:
In article <kx8Kf.3041$%14.96328@xxxxxxxxxxxxxxxxxxxxx>,
True Raptor <CB4ever@xxxxxxxxxxxxxxx> wrote:
deniz.bahar@xxxxxxxxx wrote:
My question deals with this following RULE/THEOREM:

The Substitution Rule: If u=g(x) is a differentiable function whose
range is an inteval on which f is continuous, then

integral[ f (g(x)) g' (x) ] = integral[ f (u) du ]


I can't understand why the condition "whose range is an interval on
which f is continuous" is needed. What are some cases that could
break this, and not allow us to use substitution (yet look on the
surface as a good place to use substit.) ?


If g is defined for some values for which f is not continuous, then the
integral does not exist.

May fail to exist.

Integration requires continuity.

No, it does not require continuity. Continuity guarantees the
existence of the integral, but is not required. A step function, for
example, is easily shown to be integrable even if it is not exontinuous.



so 'u = g(x)' doesn't need to be defined for a range where f is purely
continuous? I guess the book went overboard by mentioning this
condition.

Thanks

.

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